Thin Film Interference Filter and Bootstrap Method for Interference Filter Thin Film Deposition Process Control

ABSTRACT

A thin film interference filter system includes a plurality of stacked films having a determined reflectance; a modeled monitor curve; and a topmost layer configured to exhibit a wavelength corresponding to one of the determined reflectance or the modeled monitor curve. The topmost layer is placed on the plurality of stacked films and can be a low-index film such as silica or a high index film such as niobia.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims benefit of U.S. Provisional Patent Application,Ser. No. 60/609,406, entitled “Bootstrap Method for Interference FilterThin Film Deposition Process Control,” filed Sep. 13, 2004.

STATEMENT OF FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

Research described in this application was sponsored by the UnitedStates Air Force Research Laboratory grant number F22615-00-2-6059.

FIELD OF THE INVENTION

The present invention relates to thin film optical devices. Moreparticularly, the invention relates to complex interference filters.

BACKGROUND OF THE INVENTION

Highly accurate optical interference filters can be manufactured usingthin film deposition processes. These optical interference filters areused for multivariate optical computing, multiple-band-pass, and thelike and can exhibit complex optical spectra defined over a range ofwavelengths. These filters are typically constructed by depositingalternating layers of transparent materials where one layer possesses amuch larger refractive index relative to the other layer. Theoretically,the proper choice of composition, thickness and quantity of layers couldresult in a device with any desired transmission spectrum.

Among the simplest devices is the single cavity bandpass filter; i.e.,the thin-film form of an etalon. This device consists of three sets oflayers. The first stack is a dielectric mirror, a next thicker layerforms a spacer, and a second stack forms another dielectric mirror. Themirror stacks are typically fabricated by depositing alternatingtransparent materials that have an optical thickness that is one quarterof the optical wavelength of light.

To achieve theoretical optical performance, each layer must possess aprecise and specific physical thickness and refractive index. Anynonuniformity in the deposition of the layers can affect the spectralplacement and transmission or reflection characteristics of the device.A design that requires very tight manufacturing tolerances over largesubstrate areas could result in the costly rejection of many devices.Given these manufacturing limits, it would be desirable to analyze thedevices after construction and alter the devices that do not meet apredetermined optical transmission or reflection specification by someelectrical or mechanical means. For example, if the peak transmissionwavelength of a manufactured optical bandpass cavity filter was slightlyout of tolerance, it would be desirable to have a mechanism or processfor shifting the peak back to the desired spectral location. It is alsodesirable that the optical filters have precise rejection bands andpassbands that are electrically or mechanically selectable.

Mechanical methods of achieving a variable transmission spectrum deviceare well known. This includes changing a prism or grating angle, oraltering the optical spacing between mirrors of an etalon. To overcomethe performance, size and cost disadvantages of using mechanicalschemes, many have conceived of electrical methods for varying atransmission spectrum. For example, U.S. Pat. No. 5,150,236, issued Sep.22, 1992 to Patel, discloses a tunable liquid crystal etalon filter. Theliquid crystal fills the space between dielectric mirrors. Electrodes onthe mirrors are used to apply an electric field, which changes theorientation of the liquid crystal that changes the optical length fortuning. The change in the optical length corresponds to a change in thelocation of the passband. U.S. Pat. No. 5,103,340, issued Apr. 7, 1992to Dono et al., discloses piezoelectric elements placed outside theoptical path that are used to change the spacing between cascaded cavityfilters. Furthermore, U.S. Pat. No. 5,799,231, issued Aug. 25, 1998 toGates et al., discloses a variable index distributed mirror. This is adielectric mirror with half of the layers having a variable refractiveindex that is matched to other layers. Changing the applied fieldincreases the index difference that increases the reflectance. Themathematics that describes the transmission characteristics ofmultilayer films composed of electro-optic and dielectric materials arewell known.

Another electrically actuated thin film optical filter uses a series ofcrossed polarizers and liquid crystalline layers that allow electricalcontrols to vary the amount of polarization rotation in the liquid byapplying an electric field in such a way that some wavelengths areselectively transmitted. However, these electrically actuated thin filmoptical filters have the characteristic that the light must be polarizedand that the frequencies of light not passed are absorbed, notreflected. Another electrically actuated thin film optical device is thetunable liquid crystal etalon optical filter. The tunable liquid crystaletalon optical filter uses a liquid crystal between two dielectricmirrors.

The common cavity filter, such as the etalon optical filter, is anoptical filter with one or more spacer layers that are deposited in thestack and define the wavelength of the rejection and pass bands. Theoptical thickness of the film defines the placement of the passband.U.S. Pat. No. 5,710,655, issued Jan. 20, 1998 to Rumbaugh et al.,discloses a cavity thickness compensated etalon filter.

In the tunable liquid crystal etalon optical filter, an electric fieldis applied to the liquid crystal that changes the optical length betweenthe two mirrors so as to change the passband of the etalon. Stillanother tunable optical filter device tunes the passband by usingpiezoelectric elements to mechanically change the physical spacingbetween mirrors of an etalon filter.

Bulk dielectrics are made by subtractive methods like polishing from alarger piece; whereas thin film layer are made by additive methods likevapor or liquid phase deposition. A bulk optical dielectric, e.g.,greater than ten microns, disposed between metal or dielectric mirrorssuffers from excessive manufacturing tolerances and costs. Moreover, thebulk material provides unpredictable, imprecise, irregular, or otherwiseundesirable passbands. These electrical and mechanical optical filtersdisadvantageously do not provide precise rejection bands and passbandsthat are repeatably manufactured.

In an attempt to avoid some of the foregoing problems, modeling ofinterference filters can be conducted during on-line fabrication within-situ optical spectroscopy of the filter during deposition. Thecurrent state of the art for on-line correction of the depositioninvolves fitting the observed spectra to a multilayer model composed of“ideal” films based on a model for each film. The resulting modelspectra are approximations of the actual spectra. To use reflectance asan example: the measured reflectance of a stack of films can beapproximately matched to a theoretical reflectance spectrum by modeling.Layers remaining to be deposited can then be adjusted to compensate forerrors in the film stack already deposited, provided the film stack hasbeen accurately modeled. However, films vary in ways that cannot bereadily modelled using any fixed or simple physical model.Heterogeneities in the films that cannot be predicted or compensated bythis method cause the observed spectra to deviate more and more from themodel. This makes continued automatic deposition very difficult; complexfilm stacks are therefore very operator-intensive and have a highfailure rate. To improve efficiency in fabrication, laboratories thatfabricate these stacks strive to make their films as perfectly aspossible so the models are as accurate as possible.

As outlined above, many thin films are usually designed in a stack toproduce complex spectra and small variations in deposition conditionsmake it difficult to accurately model in situ film spectra for feedbackcontrol of a continuous deposition process because it is practicallyimpossible to obtain full knowledge of the detailed structure of thestack from reflectance, transmittance, ellipsometry, mass balance orother methods. Thus, a thin film interference filter is needed that isless difficult to manufacture, which will address varying refractiveindices of thin films and varying absorptions with depositionparameters.

BRIEF SUMMARY OF INVENTION

In general, the present invention is directed to a layered, thin filminterference filter and related bootstrap methods. A bootstrap methodaccording to one aspect of the invention permits a user to focus on asingle layer of a film stack as the layer is deposited to obtain anestimate of the properties of the stack. Although the single layer modelis a guideline and not a basis for compensating errors, only themost-recently-deposited layer—and not the already-deposited filmstack—need be modeled according to an aspect of the present invention.Thus, the user can neglect deviations of the stack from ideality for allother layers. The single-layer model can then be fit exactly to theobserved spectra of the film stack at each stage of deposition to allowaccurate updating of the remaining film stack for continued deposition.

The present invention works with any type of films, whether absorbing ornon-absorbing, and regardless of whether the control of the depositionconditions are state-of-the-art or not. The methods of the invention arerelatively straightforward, and a resultant thin film interferencefilter is economical to produce and use.

According to a particular aspect of the invention, a method usingexperimental measurements to determine reflectance phase and complexreflectance for arbitrary thin film stacks includes the steps ofdetermining reflectance of a stack of a plurality of films beforedepositing a topmost layer; considering a modeled monitor curve for awavelength of a high-index layer; and discarding a plurality of monitorcurves without maxima in their reflectance during the topmost layerdeposition. In this aspect of the invention, the topmost layer can be aniobia layer.

The exemplary method can also include the steps of determining ananticipated standard deviation in φ_(k) for a plurality of monitorwavelengths in the niobia layer and discarding any with σ greater than0.9 degrees. Another step in this aspect is computing expected error inδ for wavelengths with σ less than 0.9 degrees at a target thickness ofthe niobia layer. When no wavelengths have an error less than 0.9degrees, a further step according to this method is to proceed with afull model deposition of the niobia layer. When there are no maxima inthe reflectance of each of the plurality of monitor curves, another stepaccording to the exemplary method is to use only the modeled monitorcurve during the topmost layer deposition.

The method according to this aspect of the invention can also includethe step of computing a value of δ for all wavelengths based on a valuecalculated for the monitor wavelength.

The method according to this aspect of the invention can further includethe step of computing two possible values of phase angle for eachwavelength other than the monitor wavelength.

Additional steps according to the exemplary method include usinginformation extracted from the model for r_(k) at each wavelength andthe computed best value of δ, and computing an estimated standarddeviation of phase at all wavelengths except the monitor.

The method may further include the steps of using the computed phaseclosest to the model phase for r_(k) at each wavelength, measured R_(f)and R_(k) values and the computed best value of δ, and computing theestimated standard deviation of phase at all wavelengths for which themagnitude of r_(k) was estimated other than the monitor.

Further steps according to the exemplary method include the steps ofdetermining if a phase error estimate is less than about 1.3 degrees andaveraging calculated and modeled reflectance and phase values to obtaina new value for use in subsequent modeling at that wavelength.

In yet another aspect according to the exemplary method, the topmostlayer can be a silica film. Accordingly, the method can include the stepof replacing the magnitude of the amplitude reflectance at eachwavelength with √{square root over (R_(k))} whenever measuring a latestdepositing silica film having an intensity reflectance greater than 9%.The method can also include the step of determining if a phase errorestimate is less than about 1.3 degrees when the magnitude of theamplitude reflectance at each wavelength has been replaced with √{squareroot over (R_(k))} and averaging calculated and modeled reflectance andphase values to obtain a new value for use in subsequent modeling atthat wavelength.

In yet another aspect of the invention, a method for correcting thinfilm stack calculations for accurate deposition of complex opticalfilters can include the steps of determining phase angle φ_(k) at amonitor wavelength from |r′_(k)| and Rk using a first equation expressedas:${{\cos\quad\left( {\pm \quad\phi_{k}} \right)} = \frac{{{r_{k}^{\prime}}^{2}\left( {1 + {R_{k}{r_{2}}^{2}}} \right)} - {r_{2}}^{2} - R_{k}}{2\quad{r_{2}}\sqrt{R_{k}}\left( {1 - {r_{k}^{\prime}}^{2}} \right)}};$

and estimating r′_(k) using a second equation${r_{k}^{\prime} = \frac{r_{k} - r_{2}}{1 - {r_{2}r_{k}}}};$and obtaining a value for phase at a monitor wavelength.

Still another aspect of the invention includes a method for automateddeposition of complex optical interference filters including the stepsof determining from a measurement of intensity reflectance at a topmostinterface a phase angle φ at an interface k according to an equationexpressed as:${\cos\quad\left( \phi_{k} \right)} = \left( \frac{{{A\left( {1 + r_{2}^{2}} \right)}\sin\quad(\delta)} \pm {B\quad{\cos(\delta)}}}{C} \right)$$\begin{matrix}{A = {R_{f} + {r_{2}^{4}\left( {R_{f} - R_{k}} \right)} - R_{k} +}} \\{2\quad{r_{\quad 2}^{\quad 2}\left( {{\left( {1 - R_{\quad f}} \right)\left( {1 + R_{\quad k}} \right)\cos\quad\left( {2\quad\delta} \right)} - \left( {1 - {R_{\quad f}R_{\quad k}}} \right)} \right)}}\end{matrix}$$B = \sqrt{{D\left( {1 + r_{2}^{12}} \right)} + {F\left( {r_{2}^{2} + r_{2}^{10}} \right)} + {G\left( {r_{2}^{4} + r_{2}^{8}} \right)} + {Hr}_{2}^{6}}$C = sin   (δ)(4  r₂(1 − R_(f))R_(k)^(1/2)(2  r₂²cos   (2  δ) − 1 − r₂⁴))D = −(R_(f) − R_(k))² $F = {2\begin{pmatrix}{{R_{k}\left( {2 + R_{k}} \right)} + {R_{f}^{2}\left( {1 + {2\quad R_{k}}} \right)} + {2\quad{R_{f}\left( {1 - {5\quad R_{k}} + R_{k}^{2}} \right)}} -} \\{2\left( {1 - R_{f}} \right)\left( {1 - R_{k}} \right)\left( {R_{f} + R_{k}} \right)\cos\quad\left( {2\quad\delta} \right)}\end{pmatrix}}$ $\begin{matrix}{G = {{- 6} - {4\quad R_{f}} - {5\quad R_{f}^{2}} - {4\quad R_{k}} + {38\quad R_{f}R_{k}} - {4\quad R_{f}^{2}R_{k}} - {5\quad R_{k}^{2}} -}} \\{{4\quad R_{f}R_{k}^{2}} - {6\quad R_{f}^{2}R_{k}^{2}} + {8\left( {1 - R_{f}^{2}} \right)\left( {1 - R_{k}^{2}} \right)\cos\quad\left( {2\quad\delta} \right)} -} \\{2\left( {1 - R_{f}} \right)^{2}\left( {1 - R_{k}} \right)^{2}\cos\quad\left( {4\quad\delta} \right)}\end{matrix}$ $H = {4\begin{pmatrix}\begin{matrix}{3 + {2\quad R_{f}^{2}} - {10\quad R_{f}R_{k}} + {2\quad R_{k}^{2}} + {3\quad R_{f}^{2}R_{k}^{2}} -} \\{{2\left( {1 - R_{\quad f}} \right)\left( {1 - R_{\quad k}} \right)\left( {2 + R_{f} + R_{k} + {2\quad R_{f}R_{k}}} \right)\cos\quad\left( {2\quad\delta} \right)} +}\end{matrix} \\{\left( {1 - R_{f}} \right)^{2}\left( {1 - R_{k}} \right)^{2}\cos\quad\left( {4\quad\delta} \right)}\end{pmatrix}}$

According to this exemplary method, a process control for a depositionsystem is bootstrapped by detaching the deposition system from all butthe topmost interface.

The method can further include the step of validating two resultantsolutions according to the expression:$R_{f} = \frac{{2\quad r_{2}^{2}} + {R_{k}\left( {1 + r_{2}^{4}} \right)} + {2\quad r_{2}Q}}{1 + r_{2}^{4} + {2\quad r_{2}^{2}R_{k}} + {2\quad r_{2}Q}}$$\begin{matrix}{Q = {{r_{2}^{2}R_{k}^{1/2}\cos\quad\left( {{2\quad\delta} + \phi_{k}} \right)} + {R_{k}^{1/2}\cos\quad\left( {{2\quad\delta} - \phi_{k}} \right)} -}} \\{{{r_{2}\left( {1 + R_{k}} \right)}\cos\quad\left( {2\quad\delta} \right)} - {\left( {1 + r_{2}^{2}} \right)R_{k}^{1/2}\cos\quad\left( \phi_{k} \right)}}\end{matrix}$

The method can also include the step of averaging calculated and modeledreflectance and phase values to obtain a new value to be used in allfuture modeling at a given wavelength.

According to another aspect of the invention, a thin film interferencefilter system includes a plurality of stacked films having a determinedreflectance; a modeled monitor curve; and a topmost layer configured toexhibit a wavelength corresponding to one of the determined reflectanceor the modeled monitor curve, the topmost layer being disposed on theplurality of stacked films. The topmost layer according to this aspectcan be a low-index film such as silica or a high index film such asniobia.

Other aspects and advantages of the invention will be apparent from thefollowing description and the attached drawings, or can be learnedthrough practice of the invention.

BRIEF DESCRIPTION OF THE DRAWINGS

A full and enabling disclosure of the present invention, including thebest mode thereof to one of ordinary skill in the art, is set forth moreparticularly in the remainder of the specification, including referenceto the accompanying figures in which:

FIG. 1 is a schematic, cross sectional view of a stack of filmsaccording to an aspect of the invention;

FIG. 2 is a schematic, cross sectional view of a stack of films similarto FIG. 1 showing an internal interface in the stack of films accordingto another aspect of the invention;

FIG. 3 is a schematic, cross sectional view of a stack of filmsaccording to another aspect of the invention similar to FIG. 1 butshowing amplitude reflectance of a buried interface;

FIG. 4 is a representation of regions of permissible values of Rk andRmax, particularly showing lowest values of σ_(φ)k for silica accordingto an aspect of the invention;

FIG. 5 is similar to FIG. 4 but for niobia according to another aspectof the invention;

FIG. 6 is a histogram for silica as in FIG. 4;

FIG. 7 is a histogram for niobia as in FIG. 5;

FIG. 8 is a perspective plot showing an error logarithm versus Rmax andδ in accordance with an aspect of the invention; and

FIG. 9 is a histogram of FIG. 8 data.

DETAILED DESCRIPTION OF THE INVENTION

Detailed reference will now be made to the drawings in which examplesembodying the present invention are shown. Repeat use of referencecharacters in the drawings and detailed description is intended torepresent like or analogous features or elements of the presentinvention.

The drawings and detailed description provide a full and detailedwritten description of the invention and the manner and process ofmaking and using it, so as to enable one skilled in the pertinent art tomake and use it. The drawings and detailed description also provide thebest mode of carrying out the invention. However, the examples set forthherein are provided by way of explanation of the invention and are notmeant as limitations of the invention. The present invention thusincludes modifications and variations of the following examples as comewithin the scope of the appended claims and their equivalents.

Turning now to the figures, FIG. 1 shows a thin film interference filter10, which broadly includes a substrate 12 upon which a stack of films14A-X is deposited (where x represents a theoretically infinite numberof film layers). As shown, the last (alternatively, final, top ortopmost) deposited film is designated by the alphanumeral 14A whilepreviously deposited or lower level films are designated 14B-X. Anincoming ray 18 is shown in FIG. 1 being reflected at an interface 16(also referred to herein as top or top surface and when mathematicallyreferenced as k). The reflected ray is designated by the number 20. Forsimplicity, any contribution from multiple incoherent reflections in thesubstrate 12 is ignored in the following discussion and only reflectionswith respect to the film stack 14A-X are described.

Typically, reflectance of the top surface 16 is obtained using a matrixcalculation that is in turn built from the characteristic matrices ofeach of the preceding films 14A-X. As shown in FIG. 1, a computed valueof an electric field amplitude reflectance r_(k) is obtained byoptimizing the thickness of the topmost film 14A to provide the best fitover the fall spectrum consistent with an understanding of the existingfilm stack 14B-X. The calculated value of r_(k) can be considered anestimate of the actual value of r_(k) exhibited by the film stack 14A-X.Since r_(k) is a complex value, it cannot be measured directly.

Normal Matrx-type Calculations:

For standard calculations, the complex reflectance of a stack of filmsis computed using the admittances of the incident medium (often air),and the first interface of the stack. For s and p polarization, thisreflectance is: $\begin{matrix}{{r_{s}^{(f_{1})} = \frac{{- \eta_{s}^{({inc})}} - \eta_{s}^{(f_{1})}}{{- \eta_{s}^{({inc})}} + \eta_{s}^{(f_{1})}}}{r_{p}^{(f_{1})} = \frac{\eta_{p}^{({inc})} - \eta_{p}^{(f_{1})}}{\eta_{p}^{({inc})} + \eta_{p}^{(f_{1})}}}} & 6.\end{matrix}$where the subscript s or p indicates for s or p-polarized light, η is acomplex admittance, the superscript “inc” indicates the incident mediumand (f₁) indicates the first interface the light strikes when comingfrom the incident medium.

For s and p polarized light, the admittances of the incident media arewritten: $\begin{matrix}{{\eta_{s}^{({inc})} = \sqrt{ɛ_{inc} - {ɛ_{inc}{\sin\quad}^{2}\left( \theta_{inc} \right)}}}{\eta_{p}^{({inc})} = \frac{ɛ_{inc}}{\sqrt{\quad{ɛ_{inc} - {ɛ_{inc}\quad\sin^{2}\quad\left( \quad\theta_{inc} \right)}}}}}} & 7.\end{matrix}$where ∈ indicates a complex dielectric constant, and θ_(inc) is theangle of incidence in the medium of incidence.

The problematic part of the calculation is how to express the admittanceof the initial interface. The matrix calculation proceeds by relatingthe admittance of the initial interface to that of the second interface,the admittance of the second to the third, etcetera, through a series of2×2 matrices, until the calculation is related to the final interface.At the final interface, the admittance (ratio of magnetic to electricfields) is equal to the admittance of the exit medium, which is simpleto compute because there is only a single ray (the transmitted ray),rather than rays propagating in two different directions.

For a single layer stack the admittance of the initial interface isrelated to the exit medium admittance according to the followingequations: $\begin{matrix}\begin{matrix}{\begin{pmatrix}E_{x}^{(f_{1})} \\H_{y}^{(f_{1})}\end{pmatrix} = {\begin{pmatrix}s_{11} & s_{12} \\s_{21} & s_{22}\end{pmatrix}\begin{pmatrix}E_{x}^{(f_{2})} \\H_{y}^{(f_{2})}\end{pmatrix}}} \\{= {{\begin{pmatrix}s_{11} & s_{12} \\s_{21} & s_{22}\end{pmatrix}\begin{pmatrix}E_{x}^{(3)} \\H_{y}^{(3)}\end{pmatrix}}\therefore\begin{pmatrix}{E_{x}^{(f_{1})}/E_{x}^{(3)}} \\{H_{y}^{(f_{1})}/E_{x}^{(3)}}\end{pmatrix}}} \\{= {{\begin{pmatrix}s_{11} & s_{12} \\s_{21} & s_{22}\end{pmatrix}\begin{pmatrix}1 \\{- \eta_{s}^{(3)}}\end{pmatrix}}\therefore\eta_{s}^{(f_{1})}}} \\{= {\frac{H_{y}^{(f_{1})}}{E_{x}^{(f_{1})}} = \frac{s_{11} + {s_{12}\left( {- \eta_{s}^{(3)}} \right)}}{s_{21} + {s_{22}\left( {- \eta_{s}^{(3)}} \right)}}}}\end{matrix} & 8.\end{matrix}$

In these equations, the superscript (3) indicates the exit medium. Theadmittance of the exit medium for s-polarized light is given by η_(s)⁽³⁾=√{square root over (∈₃−∈_(inc) sin²(θ_(inc)))}. The negative sign infront of η_(s) ⁽³⁾ in the second line results from defining light aspropagating in the negative z direction. For s-polarized light, themagnetic and electric fields are of opposite signs in this case. Forp-polarized light, they are of the same sign. The 2×2 matrix for thesingle film is described below.

For p-polarized light, the admittance of the initial interface isarrived at in the same way: $\begin{matrix}\begin{matrix}{\begin{pmatrix}E_{y}^{(f_{1})} \\H_{x}^{(f_{1})}\end{pmatrix} = {\begin{pmatrix}p_{11} & p_{12} \\p_{21} & p_{22}\end{pmatrix}\begin{pmatrix}E_{y}^{(f_{2})} \\H_{x}^{(f_{2})}\end{pmatrix}}} \\{= {{\begin{pmatrix}p_{11} & p_{12} \\p_{21} & p_{22}\end{pmatrix}\begin{pmatrix}E_{y}^{(3)} \\H_{x}^{(3)}\end{pmatrix}}\therefore\begin{pmatrix}{E_{y}^{(f_{1})}/E_{y}^{(3)}} \\{H_{x}^{(f_{1})}/E_{y}^{(3)}}\end{pmatrix}}} \\{= {{\begin{pmatrix}p_{11} & p_{12} \\p_{21} & p_{22}\end{pmatrix}\begin{pmatrix}1 \\\eta_{p}^{(3)}\end{pmatrix}}\therefore\eta_{p}^{(f_{1})}}} \\{= {\frac{H_{x}^{(f_{1})}}{E_{y}^{(f_{1})}} = \frac{p_{11} + {p_{12}\eta_{p}^{(3)}}}{p_{21} + {p_{22}\eta_{p}^{(3)}}}}}\end{matrix} & 9.\end{matrix}$

For p-polarized light, the admittance of the exit medium is written asη_(p) ⁽³⁾=∈₃/√{square root over (∈₃−∈_(inc) sin²(θ_(inc)))}. The 2×2matrices for s and p polarizations are defined by: $\begin{matrix}{{s_{12} = {s_{22} = {p_{11} = {p_{22} = {\cos\quad\left( \delta_{film} \right)}}}}}{s_{12} = {\frac{- i}{\eta_{s}^{({film})}}\sin\quad\left( \delta_{film} \right)}}{s_{21} = {{- i}\quad\eta_{s}^{({film})}\sin\quad\left( \delta_{film} \right)}}{p_{12} = {\frac{i}{\eta_{p}^{({film})}}\sin\quad\left( \delta_{film} \right)}}{p_{21} = {i\quad\eta_{p}^{({film})}\sin\quad\left( \delta_{film} \right)}}} & 10.\end{matrix}$

In these equations, the admittances of the film are written in the sameform as the admittances of the exit medium given above, but with ∈filmreplacing ∈3. The value δfilm is the phase thickness of the film, givenby $\begin{matrix}{\delta_{film} = \frac{2\quad\pi\quad d_{film}\sqrt{ɛ_{film} - {ɛ_{inc}{\sin^{2}\left( \theta_{inc} \right)}}}}{\lambda_{0}}} & 11.\end{matrix}$where dfilm is the physical thickness of the film and λ0 is thefree-space wavelength of the incident light.

If there are multiple films, the matrix for a stack of films is obtainedfrom $\begin{matrix}{S = {\begin{pmatrix}s_{11} & s_{12} \\s_{21} & s_{22}\end{pmatrix} = {\prod\limits_{inc}^{exit}S_{films}}}} & 12.\end{matrix}$where the product is over the 2×2 matrices of each individual film fromthe entrance to the exit. The final product matrix is used as though itdescribed a single equivalent layer.

Notably in the preceding calculation, the matrices describing the filmare used as “transfer” matrices. This permits propagation of thecalculation of the admittance of the initial interface down through astack of films. The downward propagation is stopped at the substratebecause, once there are no longer rays propagating in both directions, asimple form (the admittance of the exit medium) can be written. Thus,the matrices allow an impossible calculation to be related to asimplified calculation via a 2×2 matrix.

A critical piece of understanding results from the following discussion.Referring to Equation 8 above for s polarization, −η_(s) ⁽³⁾, is theadmittance of the final interface (thus the basis for preserving thenegative sign). In comparison, p polarization, η_(p) ⁽³⁾, in Equation 6is the admittance of the final interface. Thus, both the s and pcalculation of admittance for the initial interface can be written in ageneric form: $\begin{matrix}{\eta^{(f_{1})} = {\frac{H^{(f_{1})}}{E^{(f_{1})}} = \frac{m_{11} + {m_{12}\eta^{(f_{\omega})}}}{m_{21} + {m_{22}\eta^{(f_{\omega})}}}}} & 13.\end{matrix}$where the matrix elements are for either the s or p matrices, andη^((fω)) is the admittance of the final interface. The final interfaceis always chosen because a simple expression for its admittance can bewritten in terms of the admittance of the exit medium.

A Bootstrap Method according to an aspect of the invention depends onfinding experimental values for the complex reflectance at a giveninterface in a film stack. Thus, an initial matter of using theamplitude reflectance of a film surface alone to complete the matrixcalculation for the films above the film surface in question will bedescribed.

Solving the Top of the Stack

Turning now to a problem illustrated in FIG. 2, reflectance at someinterface 160 (mathematically, k′) below a film stack 115A-X is assumed.Earlier layers 114A-X are shown “grayed out” and with diagonal lines toindicate a vague idea of what those layers 114A-X are. The reflectanceof the “known” layer 114A differs in value from the reflectancedescribed above (thus, a “prime” symbol on k at the interface 116indicates the different value). However, it is still desirable to beable to compute the reflectance spectrum of the stack of films 115A-Xlayered on top of the “known” layer 114A.

It is possible to compute the spectrum of a film stack when theamplitude reflectance at one interface at the bottom is known. Toappreciate how this calculation is done, it is useful to review theknown method for calculating reflectance and understand what theassumptions are.

If the amplitude reflectivity for an interface k (which can be anyinterface, including the final interface) is known, an equivalentexpression to Equation 13 can be obtained in terms of the admittance ofthat interface in lieu of carrying the calculation all the way down tothe substrate. The admittance for the k^(th) interface can be written asfollows: $\begin{matrix}{{{for}\quad s\text{:}\begin{Bmatrix}{\begin{pmatrix}E_{x}^{(f_{k})} \\H_{y}^{(f_{k})}\end{pmatrix} = \begin{pmatrix}{\left( {r_{k}^{\prime} + 1} \right)E_{x}^{(i_{2})}} \\{\left( {r_{k}^{\prime} - 1} \right)\eta_{s}^{(2)}{\hat{E}}_{x}^{(i_{2})}}\end{pmatrix}} \\{\eta_{s}^{(f_{k})} = {\eta_{s}^{(2)}\frac{\left( {r_{k}^{\prime} - 1} \right)}{\left( {r_{k}^{\prime} + 1} \right)}}}\end{Bmatrix}}{for}\quad p\text{:}\begin{Bmatrix}{\begin{pmatrix}E_{y}^{(f_{k})} \\H_{x}^{(f_{k})}\end{pmatrix} = \begin{pmatrix}{\left( {1 + r_{k}^{\prime}} \right)E_{y}^{(i_{2})}} \\{\left( {1 - r_{k}^{\prime}} \right)\eta_{p}^{(2)}E_{y}^{(i_{2})}}\end{pmatrix}} \\{\eta_{p}^{(f_{k})} = {\eta_{p}^{(2)}\frac{\left( {1 + r_{k}^{\prime}} \right)}{\left( {1 - r_{k}^{\prime}} \right)}}}\end{Bmatrix}} & 14.\end{matrix}$

In this expression for the admittance of the known interface, “2”indicates the admittance of the film deposited directly on the knowninterface. This can then used as the starting point for computing thereflectance of a stack of films above the known interface. Theadmittance of the top interface can be written as: $\begin{matrix}{{\eta_{s}^{(f_{1})} = {\frac{H_{y}^{(f_{1})}}{{\hat{E}}_{x}^{(f_{1})}} = {\frac{s_{21}^{\prime} + {s_{22}^{\prime}\eta_{s}^{(f_{k})}}}{s_{11}^{\prime} + {s_{12}^{\prime}\eta_{s}^{(f_{k})}}} = \frac{s_{21}^{\prime} + {s_{22}^{\prime}{\eta_{s}^{(2)}\left( \frac{r_{k}^{\prime} - 1}{r_{k}^{\prime} + 1} \right)}}}{s_{12}^{\prime} + {s_{12}^{\prime}{\eta_{s}^{(2)}\left( \frac{r_{k}^{\prime} - 1}{r_{k}^{\prime} + 1} \right)}}}}}}{\eta_{p}^{(f_{1})} = {\frac{H_{x}^{(f_{1})}}{{\hat{E}}_{y}^{(f_{1})}} = {\frac{p_{21}^{\prime} + {p_{22}^{\prime}\eta_{p}^{(f_{k})}}}{p_{11}^{\prime} + {p_{12}^{\prime}\eta_{p}^{(f_{k})}}} = \frac{p_{21}^{\prime} + {p_{22}^{\prime}{\eta_{p}^{(2)}\left( \frac{1 - r_{k}^{\prime}}{1 + r_{k}^{\prime}} \right)}}}{p_{11}^{\prime} + {p_{12}^{\prime}{\eta_{p}^{(2)}\left( \frac{1 - r_{k}^{\prime}}{1 + r_{k}^{\prime}} \right)}}}}}}} & 15.\end{matrix}$

In addition to the change in definition for the terminal interface ofthe calculation, another difference is the matrix elements come from amodified 2×2 matrix for the film stack. The modified matrix is computedas: $\begin{matrix}{{S^{\prime} = {\begin{pmatrix}s_{11}^{\prime} & s_{12}^{\prime} \\s_{21}^{\prime} & s_{22}^{\prime}\end{pmatrix} = {\prod\limits_{inc}^{k +}S_{films}}}}P^{\prime} = {\begin{pmatrix}p_{11}^{\prime} & p_{12}^{\prime} \\p_{21}^{\prime} & p_{22}^{\prime}\end{pmatrix} = {\prod\limits_{inc}^{k +}P_{films}}}} & 16.\end{matrix}$where the product is taken in the order of incident light penetratingthe stack as before, but the calculation ends with the film depositeddirectly onto the known interface. The symbol k+ in Equation 16 is usedto indicate that the product terminates with a layer 115A directly abovethe known interface 114A. The change is a significant one, in that the2×2 matrices for any of the films below the known interface 114A nolonger have to be computed.

Returning to Equation 6, the following can be expressed: $\begin{matrix}{{r_{s}^{(f_{1})} = \frac{\begin{matrix}{{\eta_{s}^{({inc})}{s_{11}^{\prime}\left( {1 + r_{k}^{\prime}} \right)}} - {\eta_{s}^{({inc})}\eta_{s}^{(2)}s_{12}^{\prime}\left( {1 - r_{k}^{\prime}} \right)} +} \\{{s_{21}^{\prime}\left( {1 + r_{k}^{\prime}} \right)} + {\eta_{s}^{(2)}{s_{22}^{\prime}\left( {1 - r_{k}^{\prime}} \right)}}}\end{matrix}}{\begin{matrix}{{\eta_{s}^{({inc})}{s_{11}^{\prime}\left( {1 + r_{k}^{\prime}} \right)}} - {\eta_{s}^{({inc})}\eta_{s}^{(2)}s_{12}^{\prime}\left( {1 - r_{k}^{\prime}} \right)} -} \\{{s_{21}^{\prime}\left( {1 + r_{k}^{\prime}} \right)} + {\eta_{s}^{(2)}{s_{22}^{\prime}\left( {1 - r_{k}^{\prime}} \right)}}}\end{matrix}}}r_{p}^{(f_{1})} = \frac{\begin{matrix}{{\eta_{p}^{({inc})}{p_{11}^{\prime}\left( {1 + r_{k}^{\prime}} \right)}} + {\eta_{p}^{({inc})}\eta_{p}^{(2)}{p_{12}^{\prime}\left( {1 - r_{k}^{\prime}} \right)}} -} \\{{p_{21}^{\prime}\left( {1 + r_{k}^{\prime}} \right)} - {\eta_{p}^{(2)}{p_{22}^{\prime}\left( {1 - r_{k}^{\prime}} \right)}}}\end{matrix}}{\begin{matrix}{{\eta_{p}^{({inc})}{p_{11}^{\prime}\left( {1 + r_{k}^{\prime}} \right)}} + {\eta_{p}^{({inc})}\eta_{p}^{(2)}{p_{12}^{\prime}\left( {1 - r_{k}^{\prime}} \right)}} +} \\{{p_{21}^{\prime}\left( {1 + r_{k}^{\prime}} \right)} + {\eta_{p}^{(2)}{p_{22}^{\prime}\left( {1 - r_{k}^{\prime}} \right)}}}\end{matrix}}} & 17.\end{matrix}$

Note that these equations feature a complex quantity called r′_(k),which, as mentioned above, is not the same as r_(k), the amplitudereflectance of the top of the film stack before the topmost layer wasadded. The two things are related to one another, however, as evident inthe following discussion.

Amplitude Reflectance of a Buried Interface

As discussed above, the magnitude of the reflectance of the interface116 (mathematically, k) in air can be learned by measuring its intensityreflectance but not the phase of the reflectance in the complex plane.With reference to FIG. 3, when the interface 116 (k) is covered byanother material 115, the known reflectance is changed. If how thereflectance changes cannot be computed in a simple way, then havinglearned anything about that reflectance is of no use. Fortunately, thereis a straightforward way to relate this to the amplitude reflectance ofthe interface in air.

In Optical Properties of Thin Solid Films (Dover Publications, Inc.,Mineola, USA, 1991), O. S. Heavens gives an expression for the amplitudereflectance of a film in terms of the reflectance of the two interfacesof the film: $\begin{matrix}{r_{film} = \frac{r_{top} + {r_{bot}{\mathbb{e}}^{- {\mathbb{i}2\delta}}}}{1 + {r_{top}r_{bot}{\mathbb{e}}^{- {\mathbb{i}2\delta}}}}} & 18.\end{matrix}$where δ is the optical phase change${\delta = \frac{2\pi\quad d\sqrt{ɛ_{2} - {ɛ_{inc}{\sin^{2}\left( \theta_{inc} \right)}}}}{\lambda_{0}}},$directly proportional to the physical thickness of the film. If rtop isreplaced with r2 (the Fresnel coefficient for reflectance off aninfinite slab of the film material with dielectric constant ∈2), andrbot is allowed to be r′_(k), the reflectance of the multilayer stackwhen the entrance medium is an infinite slab of film material, then thereflectance of the film's top interface can be written as:$\begin{matrix}{{r\left( f_{1} \right)} = \frac{r_{2} + {r_{k}^{\prime}{\mathbb{e}}^{- {\mathbb{i}2\delta}_{2}}}}{1 + {r_{2}r_{k}^{\prime}{\mathbb{e}}^{- {\mathbb{i}2\delta}_{2}}}}} & 19.\end{matrix}$

When the thickness of the film goes to zero, the exponential equals 1and the film reflectance must be identical to r_(k). This allows r′_(k)to be solved in terms of r_(k) as: $\begin{matrix}{r_{k}^{\prime} = \frac{r_{k} - r_{2}}{1 - {r_{2}r_{k}}}} & 20.\end{matrix}$

This provides an estimate of r′_(k) that is partially independent of thepreceding film stack, since r₂ does not depend on it at all and r_(k)has been modified, keeping only the phase determined by the film stackcalculation.

The Bootstrap Method.

In light of the foregoing introduction, a Bootstrap method for filmdeposition and refinement is described in the following sections; moreparticularly, steps to perform Bootstrap refinement of the optical modelof a thin film stack are provided as follows.

Step 1. Determine the reflectance of an existing film stack prior to thedeposition of a new layer.

The reflectance of a film stack provides some information regarding thecomplex amplitude reflectance that can be used to refine the model ofthe reflectance, and that is totally independent of any modeling. If onedoes not measure reflectance directly, it can be obtained by noting thattransmission plus reflectance for an absorption-free thin film stack isunity.

The relationship between the amplitude and intensity reflectance is thatthe intensity reflectance is the absolute square of the amplitudereflectance. Considering the amplitude reflectance for a moment, it willbe clear that it can be expressed in standard Cartesian coordinates on acomplex plane, or in complex polar coordinates: $\begin{matrix}{{r_{k} = {{a + {{\mathbb{i}}\quad b}} = {{r_{k}}{\mathbb{e}}^{{\mathbb{i}\phi}_{k}}}}}{{r_{k}} = \sqrt{a^{2} + b^{2}}}{\phi_{k} = {{\tan^{- 1}\left( \frac{b}{a} \right)}.}}} & 1\end{matrix}$

If the amplitude reflectance is expressed in polar coordinates, it isthe magnitude of the amplitude reflectance that is provided by a measureof intensity reflectance, |r_(k)|=√{square root over (R_(k))}.

It is possible at this point to replace the magnitude of the amplitudereflectivity in Equation 1, |r_(k)|, with the square root of theintensity reflectance. This can be done whenever the anticipated errorin future reflectance values due to errors in this initial measurementof R_(k) is expected to be small. How to obtain this relationship isshown in the following.

Calculating the Worst-case Future Reflectance

The standard deviation of the magnitude of the amplitude reflectance isgiven by Equation 3: $\begin{matrix}{{{r_{k}} = \sqrt{R_{k}}}{ɛ_{r_{k}} = {\frac{1}{\sqrt{R_{k}}}ɛ_{R_{k}}}}{\sigma_{r_{k}} = {\frac{\sigma_{R_{k}}}{\sqrt{R_{k}}}.}}} & 3\end{matrix}$In the worst-case scenario, Rk is a minimum (the amplitude reflection ison the real axis nearest the origin). This would make the error inmagnitude relatively larger. Further, the next layer could result inthis vector being advanced by π, crossing the real axis at the furthestpoint from the origin, producing a reflectance maximum. Again, in theworst case scenario, the maximum reflectance that could be generated asa result of the observed Rk is: $\begin{matrix}{{R_{\max}\left( \max \right)} = {\left( \frac{{\left( {1 + r_{2}^{2}} \right)R_{k}^{1/2}} - {2r_{2}}}{1 + r_{2}^{2} - {2r_{2}R_{k}^{1/2}}} \right)^{2}.}} & 4\end{matrix}$

To assure that replacing the reflectance amplitude with a measured valuedoes not affect future reflectance measurements by more than thestandard deviation of the reflectance measurement, the worst-casescenario must be known. This is: $\begin{matrix}{{\sigma_{R_{\max}}\left( \max \right)} = {\frac{\left( {r_{2}^{2} - 1} \right)^{2}\left( {1 + r_{2}^{2} - {2r_{2}R_{k}^{{- 1}/2}}} \right)}{\left( {1 + r_{2}^{2} - {2r_{2}R_{k}^{1/2}}} \right)^{3}}{\sigma_{R_{k}}.}}} & 5\end{matrix}$Solving Equation 5 for a factor of 1σ is not easy, and the result isvery complicated. However, a numerical solution is straightforward. Forsilica (r2=−0.2), this value is about Rk=0.19 or 19% reflectance. Forniobia (r2=−0.4), the value is about 9% reflectance. In other words,when about to deposit a silica layer, values of |r_(k)| should not beadjusted when the measured reflectance is less than 19%. When about todeposit a niobia layer, values should not be replaced when the measuredreflectance is less than 9%. Instead, assume the modeled reflectance ismore accurate in these cases, although it is not necessary to do thisoften. In the following sections, reasons are discussed to performbootstrapping only on high-index layers, so by extension, this step isrecommended only when a low-index layer is completed.

-   -   Step 2: Replace the magnitude of the amplitude reflectance at        each wavelength with √{square root over (R_(k))} whenever        measuring a freshly completed silica film with an intensity        reflectance greater than 9% or a low-index film with an        intensity reflectance greater than the limiting value of the        high-index material.

While the intensity measurement provides useful information (most of thetime) about the magnitude of reflectance, it unfortunately provides noinformation about the phase angle in the complex plane, φ. Much of theremainder of the present description relates to how to obtain thesephase angles in at least some circumstances.

Monitor Curves Can Give Non-Redundant Calculations of Phase

In most cases, the magnitude of r_(k) at the base of a niobia layer canbe obtained from the measured reflectance of the film stack terminatingin a fresh silica layer. The phase of the amplitude reflectance is moredifficult to ascertain, but there are two general approaches. The firstis to consider what values of phase are consistent with the final valueof reflectance after the next layer is added. To use this information,the optical thickness of the next layer must be known. This is sometimesa redundant calculation since the estimation of optical thickness isusually based on an understanding of the initial reflectance. This is,in fact, a weakness of the usual matrix modeling approach—thecalculation is somewhat redundant.

Without additional information, redundant calculation would normally bethe only option. However, monitor curves are usually recorded duringdeposition, and those curves contain all the information necessary tocompute the phase, φ_(k), without the need for redundant calculations.This involves the use of reflectance maxima in the monitor curves.

-   -   Step 3. Consider the modeled monitor curve for each wavelength        of a niobia (high-index) layer. Discard any monitor curves        without maxima in their reflectance during the niobia layer        deposition. If none meet this criterion, deposit the layer using        a pure model approach.

Based on Equation 14 above, the maxima and minima of a monitor curve canbe shown to depend solely on |r′_(k)|, the magnitude of the buriedinterface's reflectance, and not at all on its phase. The maximum andminimum reflectance during the monitor curve are given by Equation 21:$\begin{matrix}{{R_{\max} = \left( \frac{{r_{k}^{\prime}} + {r_{2}}}{1 + {{r_{2}}{r_{k}^{\prime}}}} \right)^{2}}{R_{\min} = {\left( \frac{{r_{k}^{\prime}} - {r_{2}}}{1 - {{r_{2}}{r_{k}^{\prime}}}} \right)^{2}.}}} & 21\end{matrix}$

Therefore, R_(max) or R_(min) can be used in the monitor curve to conveythe magnitude of the buried reflectance. This magnitude can be relatedto the reflectance as follows. $\begin{matrix}{{{r_{k}^{\prime}} = \frac{\sqrt{R_{\max}} - {r_{2}}}{1 - {{r_{2}}\sqrt{R_{\max}}}}}{or}{{{r_{k}^{\prime}} = \frac{\sqrt{R_{\min}} + {r_{2}}}{1 + {{r_{2}}\sqrt{R_{\min}}}}};{\frac{{r_{2}} - \sqrt{R_{\min}}}{1 - {{r_{2}}\sqrt{R_{\min}}}}.}}} & 22\end{matrix}$Thus, from a monitor curve covering at least a quarter wave at themonitor curve wavelength, |r′_(k)| can be determined.

A caveat to using these equations is as follows. The R_(min) expressionhas two failings. First, there are two possible solutions for |r′_(k)|based on R_(min) depending on whether |r′_(k)| is less than or greaterthan |r₂|. If it is less than |r₂|, the right-hand solution isappropriate. If it is greater than |r₂|, the left-hand solution isappropriate. The R_(max) expression also has two solutions in principlebut can be discarded because it provides nonphysical results. The secondproblem with the equation from R_(min) is the issue of experimentalerror. The error expected in the estimation of |r′_(k)| is related tothe error in measurement of R_(min) and R_(max) by Equation 23:$\begin{matrix}{{ɛ_{r_{k}^{\prime}} = {{\frac{\left( {1 - {r_{2}^{2}}} \right)}{2\sqrt{R_{\min}}\left( {1 + {{r_{2}}\sqrt{R_{\min}}}} \right)^{2}}ɛ_{R_{\min}}} \approx \frac{ɛ_{R_{\min}}}{2\sqrt{R_{\min}}}}}{ɛ_{r_{k}^{\prime}} = {{\frac{\left( {1 - {r_{2}^{2}}} \right)}{2\sqrt{R_{\max}}\left( {1 - {{r_{2}}\sqrt{R_{\max}}}} \right)^{2}}ɛ_{R_{\max}}} \approx {\frac{ɛ_{R_{\max}}}{2\sqrt{R_{\max}}}.}}}} & 23\end{matrix}$In other words, the error goes up as the key reflectance diminishes.Since the minimum is, by definition, smaller than the maximum, the errorexpected in estimating |r′_(k)| goes up accordingly. Thus, for bothreasons, the calculation of magnitude from a maximum reflectance (i.e.,a minimum in the transmission monitor curve) is preferred.Choosing the Best Monitor Wavelength

The phase angle φk at the monitor wavelength can be determined from|r′_(k)| and the Rk as: $\begin{matrix}{{\cos\left( {\pm \phi_{k}} \right)} = {\frac{{{r_{k}^{\prime}}^{2}\left( {1 + {R_{k}{r_{2}}^{2}}} \right)} - {r_{2}}^{2} - R_{k}}{2{r_{2}}\sqrt{R_{k}}\left( {1 - {r_{k}^{\prime}}^{2}} \right)}.}} & 25\end{matrix}$This, of course, provides 2 solutions. Once the phase angle φk fromEquation 25 is determined, r′_(k) can be computed using Equation 20 anda monitor curve can be generated if desired for comparison with theactual to help determine which solution is better. Afterwards, a valuefor phase at the monitor wavelength should have been obtained that is ascorrect as possible. It depends, of course, on accurately measuring themaximum reflectance value and Rk. Thus, not all wavelengths are createdequal as potential monitor wavelengths. A full-spectrum monitor(acquiring many monitor wavelengths) is the best solution, but if only asingle wavelength is available, then there is a systematic approach tochoosing the best.

Equation 25 depends, ultimately, on only two measurements: themeasurement of the initial reflectance and the measurement of themaximum reflectance. For those wavelengths that exhibit a maximumreflectance during deposition, these can be evaluated quantitatively aspossible monitor wavelengths.

It can be shown that the anticipated standard deviation of the phasecalculation can be written as: $\begin{matrix}{{\sigma_{\phi_{k}} = \sqrt{{A\quad\sigma_{R_{k}}^{2}} + {B\quad\sigma_{R_{\max}}^{2}}}}{A = \frac{\left( {{\left( {1 + r_{2}^{2}} \right)R_{k}} + {2{r_{2}\left( {1 + R_{k}} \right)}\sqrt{R_{\max}}} + {\left( {1 + r_{2}^{2}} \right)R_{\max}}} \right)^{2}}{4{R_{k}^{2}\begin{pmatrix}{{4r_{2}^{2}{R_{k}\left( {1 - R_{\max}} \right)}^{2}} -} \\\begin{pmatrix}{{2r_{2}\sqrt{R_{\max}}} + {\left( {1 + r_{2}^{2}} \right)R_{\max}} -} \\{R_{k}\left( {1 + r_{2}^{2} + {2r_{2}\sqrt{R_{\max}}}} \right)}\end{pmatrix}^{2}\end{pmatrix}}}}{B = {\frac{\left( {1 - R_{k}} \right)^{2}\left( {r_{2} + {\left( {1 + r_{2}^{2}} \right)\sqrt{R_{\max}}} + {r_{2}R_{\max}}} \right)^{2}}{{R_{\max}\left( {1 - R_{\max}} \right)}^{2}\begin{pmatrix}{{4r_{2}^{2}{R_{k}\left( {1 - R_{\max}} \right)}^{2}} -} \\\begin{pmatrix}{{2r_{2}\sqrt{R_{\max}}} + {\left( {1 + r_{2}^{2}} \right)R_{\max}} -} \\{R_{k}\left( {1 + r_{2}^{2} + {2r_{2}\sqrt{R_{\max}}}} \right)}\end{pmatrix}^{2}\end{pmatrix}}.}}} & 27\end{matrix}$

This equation is helpful selecting the best monitor wavelength for thepurpose of determining the phase of the amplitude reflectivity at themonitor wavelength.

FIG. 4 shows a representation of the regions of permissible values of Rkand Rmax, with a color code for the lowest values of σ_(φ)k for silica(assuming r2=−0.2 and the standard deviation of the reflectancemeasurements is 0.003). The lowest value possible under these conditionsis 0.0262 radians (1.5 degrees). The lower axis, Rmax, representspossible values of Rmax, while the left axis, Rk, gives possible valuesof Rk. Note that large regions of reflectance are not possible—there aremany combinations of Rk and Rmax that cannot coexist. On the boundariesof those disallowed regions, the error in estimating the phase angle,φk, becomes infinite.

The same plot for niobia, assuming r₂=−0.4, is given in FIG. 5. Theminimum value of error here under the same conditions is 0.00875 radians(0.5 degrees)—a much better phase calculation.

It is possible to develop a histogram of the number of combinations ofallowed Rmax and Rk that provide a specific level of error in φk. Thisis accomplished first by considering the range of possible Rmax valueswhen depositing a layer: it cannot, as FIGS. 4 and 5 illustrate, be lessthan the reflectance of the thin film material being deposited itself(e.g., Rmax for niobia in FIG. 5 cannot be less than −0.4²=0.2 as shownin a leftmost portion of FIG. 5 in dark yellow). Possible values of Rkcan be evaluated for each value of Rmax, and the values of phaseprecision those values provide can be determined using Equation 27. Thisis accomplished by dividing the phase thickness of the top layer intoincrements and computing the reflectance at each increment (this is donebecause the reflectances at the turning points are more likely thanthose in between the turning point values). In the end, a histogram ofthe resulting precision values can be formed and a determination made asto how likely each will appear for a given film material. This has beendone for silica (assuming r₂=−0.2, and reflectance standard deviationsof 0.003) and for niobia (assuming r₂=−0.4) as shown respectively inFIGS. 6 and 7.

For silica, 30% of all observed combinations will have a phase errorgiven by Equation 24 that is less than 2.4 degrees. No values less thanabout 1.5 degrees error in φk is possible for silica under theseconditions. For niobia, the same fraction will have φk errors less than0.9 degrees, as illustrated in the FIG. 7, also derived from a fullnumeric simulation.

Thus, the calculated phase error of possible monitor wavelengths willtend to be considerably better for niobia films than for silica. If alimit of 0.9 degrees phase error is placed on the monitors before thiscalculation is performed, only niobia will give possible monitorwavelengths, and 30% of all wavelengths (overall) will meet thiscriterion. On some layers, it is possible that no wavelengths will meetthis criterion, while on others many may do so.

-   -   Step 4. For the remaining possible monitor wavelengths in a        niobia layer deposition, determine the anticipated standard        deviation in φk. Discard any with cy greater than 0.9 degrees        (0.016 radians). If none remain, proceed with a pure model        deposition.

For low-index layers, bootstrapping is not recommended. For layers withlarger magnitudes of r₂ (such as niobia), the precision of the bootstrapis almost always better, but there is no guarantee that a specific layerwill include a set of R_(k) and R_(max) values anticipated to provideexcellent precision in calculating the phase φk. If no wavelength withan anticipated reflectance maximum meets this criterion, modeling aloneshould be relied upon to deposit the layer until a suitable bootstraplayer is reached.

An ingenious characteristic about this calculation is that it providesφk—and thus also φ′_(k)—that is consistent with the monitor curve and isindependent of δ, the phase thickness of the film. Once a valid solutionfor rk is determined, the valid solution for r′_(k) can be obtained.Thus, the final transmission value at the monitor curve wavelength canbe used to determine what value of δ is most accurate for the monitorwavelength.

If a wavelength meets the criterion specified above, then considerationshould be given as to whether the anticipated end of the layer will havea reflectance, Rf, suitable for estimating δ, the phase thickness of thelayer with some precision. This calculation is performed with Equation26, where δ depends on Rf and rk′. $\begin{matrix}{\delta = {\frac{\phi_{k}^{\prime}}{2} \pm {\frac{1}{2}{{\cos^{- 1}\left( \frac{R_{f} + {R_{f}{r_{2}}^{2}{r_{k}^{\prime}}^{2}} - {r_{2}}^{2} - {r_{k}^{\prime}}^{2}}{2{r_{2}}{r_{k}^{\prime}}\left( {1 - R_{f}} \right)} \right)}.}}}} & 26\end{matrix}$

This equation provides fairly unique solutions for δ that can be used tocorrect the reflectance at all wavelengths. The possible solutions for δthat are obtained can be tested against the observed monitor curve todetermine which is correct.

For a given monitor curve with a given expectation of Rmax, and with aknown value of r2, the sensitivity of δ to errors in Rf can bedetermined according to Equation 28: $\begin{matrix}{{\sigma_{\delta} = {{\frac{\left( {1 - {r_{2}^{2}}} \right)\left( {1 - R_{\max}} \right)}{2\left( {R_{f} - 1} \right)\sqrt{A}}}\sigma_{R_{f}}}}{A = {\left( {R_{\max} - R_{f}} \right){\begin{pmatrix}{{2{r_{2}^{2}}\left( {{R_{\max}R_{f}} + R_{f} - R_{\max} - 2} \right)} -} \\{{\left( {1 + {r_{2}^{4}}} \right)\left( {R_{\max} - R_{f}} \right)} +} \\{4\left( {{r_{2}} + {r_{2}^{3}}} \right)\left( {1 - R_{f}} \right)\sqrt{R_{\max}}}\end{pmatrix}.}}}} & 28\end{matrix}$

If the type of numerical analysis above using Equation 28 is repeated, aplot as shown in FIG. 8 is obtained (shown as the logarithm of error vs.Rmax and δ because otherwise the scale would be difficult to see). Thisplot is made over the entire range of possible values of Rmax between r₂² and 1 (on the receding axis) and δ angle between 0 and 2π (the frontaxis).

The data in FIG. 8 can also be rendered as a histogram as shown in FIG.9. The histogram in FIG. 9 implies that the error in phase thickness isusually satisfactory compared to the error in φk. At a cutoff of 0.9degrees error in δ, about 51% of the remaining monitor wavelengths forniobia, for instance, should be usable. Thus, given a good monitorwavelength for determining the phase angle φk, there is a good chance ofhaving one that also provides a good precision in δ.

To conserve time, and when modeling is running fairly well, it isreasonable to only select monitor wavelengths for niobia (in asilica/niobia stack), and only when they meet these two criteria(standard deviation of φk<0.9 degrees, and standard deviation of δ<0.9degrees at the end of the layer). When no monitor wavelengths meet thesecriteria, it is reasonable to proceed with a pure model matrix approachto depositing the next layer.

-   -   Step 5. Compute the expected error in δ for the remaining        wavelengths at the target thickness of the niobia layer. If no        wavelengths have an error less than 0.9 degrees, proceed with a        full model deposition of the layer. If some do meet this        criterion, select the lowest error in this category.        Reflectance of a New Film and Determination of the Old Phase at        Wavelengths Other than the Monitor Wavelength.

Why bother with determining the φk and δ from a monitor curve asprecisely as possible? First, if a monitor wavelength for bootstrappinghas been selected successfully, all connection to the previousdependence on the matrix calculation for the monitor wavelength may beavoided. For all other wavelengths, there is at least the opportunity todetermine the magnitude of rk, but only the original modeled estimate ofphase. The question arises: how to “repair” the phases of all otherwavelengths? Since in a single-channel monitor there are no monitorcurves at those wavelengths, the phase at each wavelength cannot bedirectly obtained. (If there was spectrograph recording all wavelengthsall of the time, such as with an FTIR system, the steps provided belowwould not be needed). However, to reach this point in bootstrapping,there must be a good value for δ for the monitor wavelength, plus goodvalues of Rmax and Rk. With this, at least some of the othermeasurements can be “fixed” to accord with measurements already taken.This will restrict errors to those of a single layer at the worst forthose wavelengths where the following calculation is possible.

From δ for the monitor wavelength, the physical thickness of the layerconsistent with the modeled refractive index of the film material can beestimated. This physical thickness and the modeled refractive index ofthe film can be used to estimate δ for all other wavelengths.

-   -   Step 6. Compute the value of δ for all wavelengths based on the        value calculated for the monitor wavelength.

Returning to Equation 16 and replacing rk′ with the definition inEquation 20, the following is obtained: $\begin{matrix}{r_{f} = {\frac{{r_{2}\left( {1 - {r_{2}r_{k}}} \right)} + {\left( {r_{k} - r_{2}} \right)\left( {{\cos\left( {2\delta} \right)} - {{\mathbb{i}}\quad{\sin\left( {2\delta} \right)}}} \right)}}{1 - {r_{2}r_{k}} + {{r_{k}\left( {r_{k} - r_{2}} \right)}\left( {{\cos\left( {2\delta} \right)} - {{\mathbb{i}}\quad{\sin\left( {2\delta} \right)}}} \right)}}.}} & 29\end{matrix}$

In Equation 29, the exponential has been replaced with a trigonometricexpression using Euler's relation. If r_(k) is replaced with a+i b, aconventional expression for r_(f) can be obtained. The complex conjugateof r_(f) can be formed and the product taken of the two. This providesthe intensity reflectance of the top interface in terms of values fromthe new film, plus a and b. The following can then replace a and b:a=cos (φ_(k))√{square root over (R _(k))}b=sin (φ_(k))√{square root over (R _(k))}  (30.)where use is made of the intensity reflectance measured in vacuum forinterface k to represent the magnitude of the amplitude reflectancevector at the interface k in vacuum.

The resulting expression can be simplified as Equation 31:$\begin{matrix}{\quad{{R_{f} = \frac{{2r_{2}^{2}} + {R_{k}\left( {1 + r_{2}^{4}} \right)} + {2r_{2}Q}}{1 + r_{2}^{4} + {2r_{2}^{2}R_{k}} + {2r_{2}Q}}}{Q = {{r_{2}^{2}R_{k}^{1/2}{\cos\left( {{2\delta} + \phi_{k}} \right)}} + {R_{k}^{1/2}{\cos\left( {{2\delta} - \phi_{k}} \right)}} - {{r_{2}\left( {1 + R_{k}} \right)}{\cos\left( {2\delta} \right)}} - {\left( {1 + r_{2}^{2}} \right)R_{k}^{1/2}{{\cos\left( \phi_{k} \right)}.}}}}}} & 31\end{matrix}$

In this expression, everything is known EXCEPT the phase angle φ atinterface k, allowing it to be determined from the measurement ofintensity reflectance at the subsequent interface.

Dispensing with numerical solution methods, this expression can besolved for the Cosine of the angle: $\begin{matrix}{\quad{{{\cos\left( \phi_{k} \right)} = \left( \frac{{{A\left( {1 + r_{2}^{2}} \right)}{\sin(\delta)}} \pm {B\quad{\cos(\delta)}}}{C} \right)}{A = {R_{f} + {r_{2}^{4}\left( {R_{f} - R_{k}} \right)} - R_{k} + {2r_{2}^{2}\begin{pmatrix}{{\left( {1 - R_{f}} \right)\left( {1 + R_{k}} \right)\cos\left( {2\delta} \right)} -} \\\left( {1 - {R_{f}R_{k}}} \right)\end{pmatrix}}}}\quad{B = \sqrt{{D\left( {1 + r_{2}^{12}} \right)} + {F\left( {r_{2}^{2} + r_{2}^{10}} \right)} + {G\left( {r_{2}^{4} + r_{8}^{2}} \right)} + {H\quad r_{6}^{2}}}}\quad{C = {{\sin(\delta)}\left( {4{r_{2}\left( {1 - R_{f}} \right)}{R_{k}^{1/2}\left( {{2r_{2}^{2}{\cos\left( {2\delta} \right)}} - 1 - r_{2}^{4}} \right)}} \right)}}\quad{D = {- \left( {R_{f} - R_{k}} \right)^{2}}}\quad{F = {2\begin{pmatrix}{{R_{k}\left( {2 + R_{k}} \right)} + {R_{f}^{2}\left( {1 + {2R_{k}}} \right)} + {2{R_{f}\left( {1 - {5R_{k}} + R_{k}^{2}} \right)}} -} \\{2\left( {1 - R_{f}} \right)\left( {1 - R_{k}} \right)\left( {R_{f} + R_{k}} \right){\cos\left( {2\delta} \right)}}\end{pmatrix}}}{G = {{- 6} - {4R_{f}} - {5R_{f}^{2}} - {4R_{k}} + {38R_{f}R_{k}} - {4R_{f}^{2}R_{k}} - {5R_{k}^{2}} - {4R_{f}R_{k}^{2}} - {6R_{f}^{2}R_{k}^{2}} + {8\left( {1 - R_{f}^{2}} \right)\left( {1 - R_{k}^{2}} \right){\cos\left( {2\delta} \right)}} - {2\left( {1 - R_{f}} \right)^{2}\left( {1 - R_{k}} \right)^{2}\cos\left( {4\delta} \right)}}}\quad{H = {4\begin{pmatrix}{3 + {2R_{f}^{2}} - {10R_{f}R_{k}} + {2R_{k}^{2}} + {3R_{f}^{2}R_{k}^{2}} -} \\{{2\left( {1 - R_{f}} \right)\left( {1 - R_{k}} \right)\left( {2 + R_{f} + R_{k} + {2R_{f}R_{k}}} \right){\cos\left( {2\delta} \right)}} +} \\{\left( {1 - R_{f}} \right)^{2}\left( {1 - R_{k}} \right)^{2}{\cos\left( {4\delta} \right)}}\end{pmatrix}}}}} & 32\end{matrix}$

In principle, Equation 32 can be used to solve for the phase angle. Bydoing so, the deposition system process control is effectively“bootstrapped” by detaching the system completely from everything thatcame before the last layer. Equation 32 provides four (4) solutions forthe phase angle; two come from the +/− portion of the calculation; twomore from the fact that cosine is an even function, so positive andnegative angles both work equally well. However, only two of thesesolutions are consistent with the measured value of R_(f). Thus,solutions should be checked via Equation 31 for validity. Only twosolutions should be left after this process is complete.

-   -   Step 7. Compute the two possible values of phase angle for each        wavelength other than the monitor wavelength.        Estimating Error for Non-Monitor Wavelengths

The phase angle is dependent on three reflectivity measurements(R_(max), R_(f) and R_(k)), those being scrambled together in Equation32. While this works well for a hypothetical system with no noise, areal spectrometer exhibits errors in measurement of the intensitytransmittance.

Based on work already done, the analysis of Equation 32 is fairlystraightforward for extending phase information to other wavelengths.The following expression can be constructed from it: $\begin{matrix}{{ɛ_{\phi_{k}} = {\frac{- 1}{\sin\left( \phi_{k} \right)}\left( {{\left( \frac{\partial{\cos\left( \phi_{k} \right)}}{\partial R_{k}} \right)ɛ_{R_{k}}} + {\left( \frac{\partial{\cos\left( \phi_{k} \right)}}{\partial R_{f}} \right)ɛ_{R_{f}}} + {\left( \frac{\partial{\cos\left( \phi_{k} \right)}}{\partial\delta} \right)ɛ_{\delta}}} \right)}}\quad{\sigma_{\phi_{k}} \approx {\frac{\sigma_{R}}{{\sin\left( \phi_{k} \right)}}{\left( {\left( \frac{\partial{\cos\left( \phi_{k} \right)}}{\partial R_{k}} \right)^{2} + \left( \frac{\partial{\cos\left( \phi_{k} \right)}}{\partial R_{f}} \right)^{2}} \right)^{1/2}.}}}} & 33\end{matrix}$

In Equation 33 error in phase thickness has been omitted from thecalculation since it is of minor concern at this stage. The phasethickness was settled previously for the sake of argument; therefore,making that approximation, computing the sine of the angle and thesensitivity of the cosine functions to errors in R_(k) and R_(f) remain.There is a subtle issue to be considered at this point. What angleshould be focused on—the modeled angle or the computed angle usingEquations 31/32? In principle, both values are available.

Thus, the conservative answer to the foregoing question is “both”.Assume, for example, that the model has a complex reflectance rk at agiven wavelength from which the phase angle φk and the predicted valueof Rk can be obtained. Using a best estimate of the phase thickness atthe wavelength, Rf can be computed using Equation 31. Next, compute thephase angle using Equation 32 after varying Rk and Rf each by a smallamount—e.g., by 10⁻⁵, 1/100^(th) of a percent transmission. Since themodeled phase is known exactly, it will be trivial to identify whichresults are the ones closest to the modeled phase; therefore, thesensitivities are computed as: $\begin{matrix}{\frac{\partial{\cos\left( \phi_{k} \right)}}{\partial R_{k}} \approx \frac{{\cos\left( {\phi_{k}\left( {R_{k,m},R_{f,m},\delta} \right)} \right)} - {\cos\left( {\phi_{k}\left( {{R_{k,m} - 10^{- 5}},R_{f,m},\delta} \right)} \right)}}{10^{- 5}}} & 34 \\{\frac{\partial{\cos\left( \phi_{k} \right)}}{\partial R_{f}} \approx {\frac{{\cos\left( {\phi_{k}\left( {R_{k,m},R_{f,m},\delta} \right)} \right)} - {\cos\left( {\phi_{k}\left( {R_{k,m},{R_{f,m} - 10^{- 5}},\delta} \right)} \right)}}{10^{- 5}}.}} & \quad\end{matrix}$

From the model, the sine of the phase angle that appears in Equation 33is trivial to obtain.

Thus, the standard deviation of the phase calculation can be estimatedfrom the model. If the model were absolutely trustworthy, this would besufficient; however, this is not the case.

-   -   Step 8. Using information extracted from the model for r_(k) at        each wavelength and the computed best value of δ, compute the        estimated standard deviation of phase at all wavelengths except        the monitor.

Since the model is not completely trustworthy, the process is repeatedusing values taken from the calculated value of phase closest to themodel value of phase. In other words, for the two possible values of φk,the one closest numerically to the model phase is selected. Now, keyingon that value, compute the sensitivities according to Equation 34 usingthe measured values of R_(k), R_(f) and the estimate of δ for thewavelength being tested.

-   -   Step 9. Using the computed phase closest to the model phase for        r_(k) at each wavelength, the measured R_(f) and R_(k) values        and the computed best value of δ, compute the estimated standard        deviation of phase at all wavelengths for which the magnitude of        r_(k) was estimated (Step 2) other than the monitor.

Now, compute the estimated error in phase as follows:σ_(φk)(estimated)=(σ_(φk) ²(model)+σ_(φk) ²(calculated))^(1/2)  35.

What are the probable limits to this calculation? A bit ofexperimentation suggests that the value of standard deviation in phaseangle could be between about that of the monitor measurement (on the lowend) to nearly infinite. A sufficient approach is to replace the modeledvalues with calculated values only if a rather conservative cutoff isobtained. If the error estimates are less than about 1.3 degrees fromEquation 35 (square root of two times 0.9), and if the reflectance wasgreater than 9 percent at the start of the layer (Step 2), then themodel and calculated values are averaged together to obtain a newestimate. If either the phase error OR the reflectance criteria are notmet, the estimate is returned alone to the model.

-   -   Step 10. If the phase error estimate is less than 1.3 degrees        AND the criterion of Step 2 is met, average the calculated and        modeled reflectance and phase values to obtain a new value that        will be used in all future modeling at that wavelength.

A further refinement of this approach is to fit the phase valuesobtained above to a Kramers-Kronig model of the phase to fill in valuesof the phase that have not been determined previously.

While preferred embodiments of the invention have been shown anddescribed, those of ordinary skill in the art will recognize thatchanges and modifications may be made to the foregoing examples withoutdeparting from the scope and spirit of the invention. Furthermore, thoseof ordinary skill in the art will appreciate that the foregoingdescription is by way of example only, and is not intended to limit theinvention so further described in such appended claims. It is intendedto claim all such changes and modifications as fall within the scope ofthe appended claims and their equivalents.

1. A method using experimental measurements to determine reflectancephase and complex reflectance for arbitrary thin film stacks, the methodcomprising the steps of: determining reflectance of a stack of aplurality of films before depositing a topmost layer; considering amodeled monitor curve for a wavelength of a high-index layer; anddiscarding a plurality of monitor curves without maxima in theirreflectance during the topmost layer deposition.
 2. The method as inclaim 1, wherein the topmost layer is a niobia layer.
 3. The method asin claim 2, further comprising the steps of determining an anticipatedstandard deviation in φk for a plurality of monitor wavelengths in theniobia layer and discarding any with σ greater than 0.9 degrees.
 4. Themethod as in claim 3, further comprising the step of computing expectederror in δ for wavelengths with σ less than 0.9 degrees at a targetthickness of the niobia layer.
 5. The method as in claim 4, furthercomprising the step proceeding with a full model deposition of theniobia layer when no wavelengths have an error less than 0.9 degrees. 6.The method as in claim 1, further comprising the step of using only themodeled monitor curve during the topmost layer deposition when there isno maxima in the reflectance of each of the plurality of monitor curves.7. The method as in claim 1, further comprising the step of computing avalue of δ for all wavelengths based on a value calculated for themonitor wavelength.
 8. The method as in claim 1, further comprising thestep of computing two possible values of phase angle for each wavelengthother than the monitor wavelength.
 9. The method as in claim 1, furthercomprising the steps of using information extracted from the model forr_(k) at each wavelength and the computed best value of δ, and computingan estimated standard deviation of phase at all wavelengths except themonitor.
 10. The method as in claim 9, further comprising the steps ofusing the computed phase closest to the model phase for r_(k) at eachwavelength, measured R_(f) and R_(k) values and the computed best valueof δ, and computing the estimated standard deviation of phase at allwavelengths for which the magnitude of r_(k) was estimated other thanthe monitor.
 11. The method as in claim 9, further comprising the stepsof determining if a phase error estimate is less than about 1.3 degreesand averaging calculated and modeled reflectance and phase values toobtain a new value for use in subsequent modeling at that wavelength.12. The method as in claim 1, wherein the topmost layer is a silicafilm.
 13. The method as in claim 12, further comprising the step ofreplacing the magnitude of the amplitude reflectance at each wavelengthwith √{square root over (R_(k))} whenever measuring a latest depositingsilica film having an intensity reflectance greater than 9%.
 14. Themethod as in claim 13, further comprising the step of determining if aphase error estimate is less than about 1.3 degrees when the magnitudeof the amplitude reflectance at each wavelength has been replaced with√{square root over (R_(k))} and averaging calculated and modeledreflectance and phase values to obtain a new value for use in subsequentmodeling at that wavelength.
 15. A method for correcting thin film stackcalculations for accurate deposition of complex optical filters, themethod comprising the steps of: determining phase angle φk at a monitorwavelength from |r′_(k)| and Rk using a first equation expressed as:${{\cos\left( {\pm \phi_{k}} \right)} = \frac{{{r_{k}^{\prime}}^{2}\left( {1 + {R_{k}{r_{2}}^{2}}} \right)} - {r_{2}}^{2} - R_{k}}{2{r_{2}}\sqrt{R_{k}}\left( {1 - {r_{k}^{\prime}}^{2}} \right)}};$and estimating r′_(k) using a second equation${r_{k}^{\prime} = \frac{r_{k} - r_{2}}{1 - {r_{2}r_{k}}}};$ andobtaining a value for phase at a monitor wavelength.
 16. A method forautomated deposition of complex optical interference filters, the methodcomprising the steps of: determining from a measurement of intensityreflectance at a topmost interface a phase angle φ at an interface kaccording to an equation expressed as:$\quad{{\cos\left( \phi_{k} \right)} = \left( \frac{{{A\left( {1 + r_{2}^{2}} \right)}{\sin(\delta)}} \pm {B\quad{\cos(\delta)}}}{C} \right)}$$\quad{A = {R_{f} + {r_{2}^{4}\left( {R_{f} - R_{k}} \right)} - R_{k} + {2{r_{2}^{2}\begin{pmatrix}{{\left( {1 - R_{f}} \right)\left( {1 + R_{k}} \right)\cos\left( {2\delta} \right)} -} \\\left( {1 - {R_{f}R_{k}}} \right)\end{pmatrix}}}}}$$\quad{B = \sqrt{{D\left( {1 + r_{2}^{12}} \right)} + {F\left( {r_{2}^{2} + r_{2}^{10}} \right)} + {G\left( {r_{2}^{4} + r_{8}^{2}} \right)} + {H\quad r_{6}^{2}}}}$  C = sin (δ)(4r₂(1 − R_(f))R_(k)^(1/2)(2r₂²cos (2δ) − 1 − r₂⁴))  D = −(R_(f) − R_(k))² $\quad{F = {2\begin{pmatrix}{{R_{k}\left( {2 + R_{k}} \right)} + {R_{f}^{2}\left( {1 + {2R_{k}}} \right)} + {2{R_{f}\left( {1 - {5R_{k}} + R_{k}^{2}} \right)}} -} \\{2\left( {1 - R_{f}} \right)\left( {1 - R_{k}} \right)\left( {R_{f} + R_{k}} \right){\cos\left( {2\delta} \right)}}\end{pmatrix}}}$G = −6 − 4R_(f) − 5R_(f)² − 4R_(k) + 38R_(f)R_(k) − 4R_(f)²R_(k) − 5R_(k)² − 4R_(f)R_(k)² − 6R_(f)²R_(k)² + 8(1 − R_(f)²)(1 − R_(k)²)cos (2δ) − 2(1 − R_(f))²(1 − R_(k))²cos (4δ)$\quad{H = {4\begin{pmatrix}{3 + {2R_{f}^{2}} - {10R_{f}R_{k}} + {2R_{k}^{2}} + {3R_{f}^{2}R_{k}^{2}} -} \\{{2\left( {1 - R_{f}} \right)\left( {1 - R_{k}} \right)\left( {2 + R_{f} + R_{k} + {2R_{f}R_{k}}} \right){\cos\left( {2\delta} \right)}} +} \\{\left( {1 - R_{f}} \right)^{2}\left( {1 - R_{k}} \right)^{2}{\cos\left( {4\delta} \right)}}\end{pmatrix}}}$
 17. The method as in claim 16, wherein a processcontrol for a deposition system is bootstrapped by detaching thedeposition system from all but the topmost interface.
 18. The method asin claim 16, further comprising the step of validating two resultantsolutions according to the expression:$\quad{R_{f} = \frac{{2r_{2}^{2}} + {R_{k}\left( {1 + r_{2}^{4}} \right)} + {2r_{2}Q}}{1 + r_{2}^{4} + {2r_{2}^{2}R_{k}} + {2r_{2}Q}}}$Q = r₂²R_(k)^(1/2)cos (2δ + ϕ_(k)) + R_(k)^(1/2)cos (2δ − ϕ_(k)) − r₂(1 + R_(k))cos (2δ) − (1 + r₂²)R_(k)^(1/2)cos
 19. The method as in claim 16, further comprising the step of averagingcalculated and modeled reflectance and phase values to obtain a newvalue to be used in all future modeling at a given wavelength.
 20. Athin film interference filter system, comprising: a plurality of stackedfilms having a determined reflectance; a modeled monitor curve; and atopmost layer configured to exhibit a wavelength corresponding to one ofthe determined reflectance or the modeled monitor curve, the topmostlayer being disposed on the plurality of stacked films.
 21. The thinfilm interference filter system as in claim 20, wherein the topmostlayer is a silica film.
 22. The thin film interference filter system asin claim 20, wherein the topmost layer is a niobia film.